< , > , <= , >= , != , = , Not , And , Or , IsFreeOf , IsZeroVector , IsNonObject , IsEven , IsOdd , IsFunction , IsAtom , IsString , IsNumber , IsList , IsNumericList , IsBound , IsBoolean , IsNegativeNumber , IsNegativeInteger , IsPositiveNumber , IsPositiveInteger , IsNotZero , IsNonZeroInteger , IsInfinity , IsPositiveReal , IsNegativeReal , IsConstant .

# Predicates

A predicate is a function that returns a boolean value, i.e. `True` or `False`. Predicates are often used in patterns, For instance, a rule that only holds for a positive integer would use a pattern like `n_IsPositiveInteger`.

< | Test for "less than"
> | Test for "greater than"
<= | Test for "less or equal"
>= | Test for "greater or equal"
!= | Test for "not equal"
= | Test for equality of expressions
Not | Logical negation
And | Logical conjunction
Or | Logical disjunction
IsFreeOf | Test whether expression depends on variable
IsZeroVector | Test whether list contains only zeroes
IsNonObject | Test whether argument is not an `Object()`
IsEven | Test for an even integer
IsOdd | Test for an odd integer
IsFunction | Test for a composite object
IsAtom | Test for an atom
IsString | Test for an string
IsNumber | Test for a number
IsList | Test for a list
IsNumericList | test for a list of numbers
IsBound | test for a bound variable
IsBoolean | Test for a Boolean value
IsNegativeNumber | Test for a negative number
IsNegativeInteger | Test for a negative integer
IsPositiveNumber | Test for a positive number
IsPositiveInteger | Test for a positive integer
IsNotZero | Test for a nonzero number
IsNonZeroInteger | Test for a nonzero integer
IsInfinity | Test for an infinity
IsPositiveReal | Test for a numerically positive value
IsNegativeReal | Test for a numerically negative value
IsConstant | Test for a constant

###

< -- Test for "less than"

Standard math library

##### Calling format:

    e1 < e2 (prec. 9)

##### Parameters:

e1, e2 - expressions to be compared

##### Description:

The two expression are evaluated. If both results are numeric, they are compared. If the first expression is smaller than the second one, the result is `True` and it is `False` otherwise. If either of the expression is not numeric, after evaluation, the expression is returned with evaluated arguments.

The word "numeric" in the previous paragraph has the following meaning. An expression is numeric if it is either a number (i.e. `IsNumber` returns `True`), or the quotient of two numbers, or an infinity (i.e. `IsInfinity` returns `True`).

##### Examples:

    In> 2 < 5;
    Out> True;
    In> Cos(1) < 5;
    Out> Cos(1)<5;
    In> N(Cos(1)) < 5;
    Out> True

##### See also:

IsNumber , IsInfinity , N .

###

> -- Test for "greater than"

Standard math library

##### Calling format:

    e1 > e2 (prec. 9)

##### Parameters:

e1, e2 - expressions to be compared

##### Description:

The two expression are evaluated. If both results are numeric, they are compared. If the first expression is larger than the second one, the result is `True` and it is `False` otherwise. If either of the expression is not numeric, after evaluation, the expression is returned with evaluated arguments.

The word "numeric" in the previous paragraph has the following meaning. An expression is numeric if it is either a number (i.e. `IsNumber` returns `True`), or the quotient of two numbers, or an infinity (i.e. `IsInfinity` returns `True`).

##### Examples:

    In> 2 > 5;
    Out> False;
    In> Cos(1) > 5;
    Out> Cos(1)>5;
    In> N(Cos(1)) > 5;
    Out> False

##### See also:

IsNumber , IsInfinity , N .

###

<= -- Test for "less or equal"

Standard math library

##### Calling format:

    e1 <= e2 (prec. 9)

##### Parameters:

e1, e2 - expressions to be compared

##### Description:

The two expression are evaluated. If both results are numeric, they are compared. If the first expression is smaller than or equals the second one, the result is `True` and it is `False` otherwise. If either of the expression is not numeric, after evaluation, the expression is returned with evaluated arguments.

The word "numeric" in the previous paragraph has the following meaning. An expression is numeric if it is either a number (i.e. `IsNumber` returns `True`), or the quotient of two numbers, or an infinity (i.e. `IsInfinity` returns `True`).

##### Examples:

    In> 2 <= 5;
    Out> True;
    In> Cos(1) <= 5;
    Out> Cos(1)<=5;
    In> N(Cos(1)) <= 5;
    Out> True

##### See also:

IsNumber , IsInfinity , N .

###

>= -- Test for "greater or equal"

Standard math library

##### Calling format:

    e1 >= e2 (prec. 9)

##### Parameters:

e1, e2 - expressions to be compared

##### Description:

The two expression are evaluated. If both results are numeric, they are compared. If the first expression is larger than or equals the second one, the result is `True` and it is `False` otherwise. If either of the expression is not numeric, after evaluation, the expression is returned with evaluated arguments.

The word "numeric" in the previous paragraph has the following meaning. An expression is numeric if it is either a number (i.e. `IsNumber` returns `True`), or the quotient of two numbers, or an infinity (i.e. `IsInfinity` returns `True`).

##### Examples:

    In> 2 >= 5;
    Out> False;
    In> Cos(1) >= 5;
    Out> Cos(1)>=5;
    In> N(Cos(1)) >= 5;
    Out> False

##### See also:

IsNumber , IsInfinity , N .

###

!= -- Test for "not equal"

Standard math library

##### Calling format:

    e1 != e2 (prec. 9)

##### Parameters:

e1, e2 - expressions to be compared

##### Description:

Both expression are evaluated and compared. If they turn out to be equal, the result is `False`. Otherwise, the result is `True`.

The expression `e1 != e2` is equivalent to `Not(e1 = e2)`.

##### Examples:

    In> 1 != 2;
    Out> True;
    In> 1 != 1;
    Out> False;

##### See also:

= .

###

= -- Test for equality of expressions

Standard math library

##### Calling format:

    e1 = e2 (prec. 9)

##### Parameters:

e1, e2 - expressions to be compared

##### Description:

Both expression are evaluated and compared. If they turn out to be equal, the result is `True`. Otherwise, the result is `False`. The function `Equals` does the same.

Note that the test is on syntactic equality, not mathematical equality. Hence even if the result is `False`, the expressions can still be mathematically equal; see the examples below. Put otherwise, this function tests whether the two expressions would be displayed in the same way if they were printed.

##### Examples:

    In> e1 := (x+1) * (x-1);
    Out> (x+1)*(x-1);
    In> e2 := x^2 - 1;
    Out> x^2-1;

    In> e1 = e2;
    Out> False;
    In> Expand(e1) = e2;
    Out> True;

##### See also:

!= , Equals .

###

Not -- Logical negation

Internal function

##### Calling format:

    Not expr

##### Parameters:

`expr` -- a boolean expression

##### Description:

Not returns the logical negation of the argument expr. If `expr` is `False` it returns `True`, and if `expr` is `True`, `Not expr` returns `False`. If the argument is neither `True` nor `False`, it returns the entire expression with evaluated arguments.

##### Examples:

    In> Not True
    Out> False;
    In> Not False
    Out> True;
    In> Not(a)
    Out> Not a;

##### See also:

And , Or .

###

And -- Logical conjunction

Internal function

##### Calling format:

    a1 And a2  (prec. 100)
    And(a1, a2, a3, ..., aN)

##### Parameters:

a1, ..., aN - boolean values (may evaluate to `True` or `False`)

##### Description:

This function returns True if all arguments are true. The And operation is lazy, it returns False as soon as a False argument is found (from left to right). If an argument other than True or False is encountered a new And expression is returned with all arguments that didn't evaluate to True or False yet.

##### Examples:

    In> True And False
    Out> False;
    In> And(True,True)
    Out> True;
    In> False And a
    Out> False;
    In> True And a
    Out> And(a);
    In> And(True,a,True,b)
    Out> b And a;

##### See also:

Or , Not .

###

Or -- Logical disjunction

Internal function

##### Calling format:

    a1 Or a2  (prec. 101)
    Or(a1, a2, a3, ..., aN)

##### Parameters:

a1, ..., aN - boolean expressions (may evaluate to `True` or `False`)

##### Description:

This function returns `True` if an argument is encountered that is true (scanning from left to right). The `Or` operation is "lazy", it returns `True` as soon as a `True` argument is found (from left to right). If an argument other than `True` or `False` is encountered, an unevaluated `Or` expression is returned with all arguments that didn't evaluate to `True` or `False` yet.

##### Examples:

    In> True Or False
    Out> True;
    In> False Or a
    Out> Or(a);
    In> Or(False,a,b,True)
    Out> True;

##### See also:

And , Not .

###

IsFreeOf -- Test whether expression depends on variable

Standard math library

##### Calling format:

    IsFreeOf(expr, var)
    IsFreeOf(expr, {var, ...})

##### Parameters:

expr - expression to test

var - variable to look for in "expr"

##### Description:

This function checks whether the expression "expr" (after being evaluated) depends on the variable "var". It returns `False` if this is the case and `True` otherwise.

The second form test whether the expression depends on any of the variables named in the list. The result is `True` if none of the variables appear in the expression and `False` otherwise.

##### Examples:

    In> IsFreeOf(Sin(x), x);
    Out> False;
    In> IsFreeOf(Sin(x), y);
    Out> True;
    In> IsFreeOf(D(x) a*x+b, x);
    Out> True;
    In> IsFreeOf(Sin(x), {x,y});
    Out> False;

The third command returns `True` because the expression `D(x) a*x+b` evaluates to `a`, which does not depend on `x`.

##### See also:

Contains .

###

IsZeroVector -- Test whether list contains only zeroes

Standard math library

##### Calling format:

    IsZeroVector(list)

##### Parameters:

list - list to compare against the zero vector

##### Description:

The only argument given to `IsZeroVector` should be a list. The result is `True` if the list contains only zeroes and `False` otherwise.

##### Examples:

    In> IsZeroVector({0, x, 0});
    Out> False;
    In> IsZeroVector({x-x, 1 - D(x) x});
    Out> True;

##### See also:

IsList , ZeroVector .

###

IsNonObject -- Test whether argument is not an `Object()`

Standard math library

##### Calling format:

    IsNonObject(expr)

##### Parameters:

expr - the expression to examine

##### Description:

This function returns `True` if "expr" is not of the form `Object(...)` and `False` otherwise.

##### Bugs

In fact, the result is always `True`.

##### See also:

Object .

###

IsEven -- Test for an even integer

Standard math library

##### Calling format:

    IsEven(n)

##### Parameters:

n - integer to test

##### Description:

This function tests whether the integer "n" is even. An integer is even if it is divisible by two. Hence the even numbers are 0, 2, 4, 6, 8, 10, etcetera, and -2, -4, -6, -8, -10, etcetera.

##### Examples:

    In> IsEven(4);
    Out> True;
    In> IsEven(-1);
    Out> False;

##### See also:

IsOdd , IsInteger .

###

IsOdd -- Test for an odd integer

Standard math library

##### Calling format:

    IsOdd(n)

##### Parameters:

n - integer to test

##### Description:

This function tests whether the integer "n" is odd. An integer is odd if it is not divisible by two. Hence the odd numbers are 1, 3, 5, 7, 9, etcetera, and -1, -3, -5, -7, -9, etcetera.

##### Examples:

    In> IsOdd(4);
    Out> False;
    In> IsOdd(-1);
    Out> True;

##### See also:

IsEven , IsInteger .

###

IsFunction -- Test for a composite object

Internal function

##### Calling format:

    IsFunction(expr)

##### Parameters:

expr - expression to test

##### Description:

This function tests whether "expr" is a composite object, ie. not an atom. This includes not only obvious functions like `f(x)`, but also expressions like `x+5` and lists.

##### Examples:

    In> IsFunction(x+5);
    Out> True;
    In> IsFunction(x);
    Out> False;

##### See also:

IsAtom , IsList , Type .

###

IsAtom -- Test for an atom

Internal function

##### Calling format:

    IsAtom(expr)

##### Parameters:

expr - expression to test

##### Description:

This function tests whether "expr" is an atom. Numbers, strings, and variables are all atoms.

##### Examples:

    In> IsAtom(x+5);
    Out> Falso;
    In> IsAtom(5);
    Out> True;

##### See also:

IsFunction , IsNumber , IsString .

###

IsString -- Test for an string

Internal function

##### Calling format:

    IsString(expr)

##### Parameters:

expr - expression to test

##### Description:

This function tests whether "expr" is a string. A string is a text within quotes, eg. `"duh"`.

##### Examples:

    In> IsString("duh");
    Out> True;
    In> IsString(duh);
    Out> False;

##### See also:

IsAtom , IsNumber .

###

IsNumber -- Test for a number

Internal function

##### Calling format:

    IsNumber(expr)

##### Parameters:

expr - expression to test

##### Description:

This function tests whether "expr" is a number. There are two kinds of numbers, integers (like 6) and reals (like -2.75 or 6.0). Note that a complex number is represented by the `Complex` function, so `IsNumber` will return `False`.

##### Examples:

    In> IsNumber(6);
    Out> True;
    In> IsNumber(3.25);
    Out> True;
    In> IsNumber(I);
    Out> False;
    In> IsNumber("duh");
    Out> False;

##### See also:

IsAtom , IsString , IsInteger , IsPositiveNumber , IsNegativeNumber , Complex .

###

IsList -- Test for a list

Internal function

##### Calling format:

    IsList(expr)

##### Parameters:

expr - expression to test

##### Description:

This function tests whether "expr" is a list. A list is a sequence between curly braces, like `{2, 3, 5}`.

##### Examples:

    In> IsList({2,3,5});
    Out> True;
    In> IsList(2+3+5);
    Out> False;

##### See also:

IsFunction .

###

IsNumericList -- test for a list of numbers

Standard math library

##### Calling format:

    IsNumericList({list})

##### Parameters:

`{list}` -- a list

##### Description:

Returns `True` when called on a list of numbers or expressions that evaluate to numbers using `N()`. Returns `False` otherwise.

##### See also:

N , IsNumber .

###

IsBound -- test for a bound variable

Internal function

##### Calling format:

    IsBound(var)

##### Parameters:

var - variable to test

##### Description:

This function tests whether the variable "var" is bound, ie. whether it has been assigned a value. The argument "var" is not evaluated.

##### Examples:

    In> IsBound(x);
    Out> False;
    In> x := 5;
    Out> 5;
    In> IsBound(x);
    Out> True;

##### See also:

IsAtom .

###

IsBoolean -- Test for a Boolean value

Standard math library

##### Calling format:

    IsBoolean(expression)

##### Parameters:

expression - an expression

##### Description:

IsBoolean returns True if the argument is of a boolean type. This means it has to be either True, False, or an expression involving functions that return a boolean result, like `=`, `>`, `<`, `>=`, `<=`, `!=`, `And`, `Not`, `Or`.

##### Examples:

    In> IsBoolean(a)
    Out> False;
    In> IsBoolean(True)
    Out> True;
    In> IsBoolean(a And b)
    Out> True;

##### See also:

True , False .

###

IsNegativeNumber -- Test for a negative number

Standard math library

##### Calling format:

    IsNegativeNumber(n)

##### Parameters:

`n` - number to test

##### Description:

`IsNegativeNumber(n)` evaluates to `True` if n is (strictly) negative, i.e. if n<0. If `n` is not a number, the functions return `False`.

##### Examples:

    In> IsNegativeNumber(6);
    Out> False;
    In> IsNegativeNumber(-2.5);
    Out> True;

##### See also:

IsNumber , IsPositiveNumber , IsNotZero , IsNegativeInteger , IsNegativeReal .

###

IsNegativeInteger -- Test for a negative integer

Standard math library

##### Calling format:

    IsNegativeInteger(n)

##### Parameters:

`n` - integer to test

##### Description:

This function tests whether the integer `n` is (strictly) negative. The negative integers are -1, -2, -3, -4, -5, etcetera. If `n` is not a integer, the function returns `False`.

##### Examples:

    In> IsNegativeInteger(31);
    Out> False;
    In> IsNegativeInteger(-2);
    Out> True;

##### See also:

IsPositiveInteger , IsNonZeroInteger , IsNegativeNumber .

###

IsPositiveNumber -- Test for a positive number

Standard math library

##### Calling format:

    IsPositiveNumber(n)

##### Parameters:

`n` - number to test

##### Description:

`IsPositiveNumber(n)` evaluates to `True` if n is (strictly) positive, i.e. if n>0. If `n` is not a number the function returns `False`.

##### Examples:

    In> IsPositiveNumber(6);
    Out> True;
    In> IsPositiveNumber(-2.5);
    Out> False;

##### See also:

IsNumber , IsNegativeNumber , IsNotZero , IsPositiveInteger , IsPositiveReal .

###

IsPositiveInteger -- Test for a positive integer

Standard math library

##### Calling format:

    IsPositiveInteger(n)

##### Parameters:

`n` - integer to test

##### Description:

This function tests whether the integer `n` is (strictly) positive. The positive integers are 1, 2, 3, 4, 5, etcetera. If `n` is not a integer, the function returns `False`.

##### Examples:

    In> IsPositiveInteger(31);
    Out> True;
    In> IsPositiveInteger(-2);
    Out> False;

##### See also:

IsNegativeInteger , IsNonZeroInteger , IsPositiveNumber .

###

IsNotZero -- Test for a nonzero number

Standard math library

##### Calling format:

    IsNotZero(n)

##### Parameters:

`n` - number to test

##### Description:

`IsNotZero(n)` evaluates to `True` if `n` is not zero. In case `n` is not a number, the function returns `False`.

##### Examples:

    In> IsNotZero(3.25);
    Out> True;
    In> IsNotZero(0);
    Out> False;

##### See also:

IsNumber , IsPositiveNumber , IsNegativeNumber , IsNonZeroInteger .

###

IsNonZeroInteger -- Test for a nonzero integer

Standard math library

##### Calling format:

    IsNonZeroInteger(n)

##### Parameters:

`n` - integer to test

##### Description:

This function tests whether the integer `n` is not zero. If `n` is not an integer, the result is `False`.

##### Examples:

    In> IsNonZeroInteger(0)
    Out> False;
    In> IsNonZeroInteger(-2)
    Out> True;

##### See also:

IsPositiveInteger , IsNegativeInteger , IsNotZero .

###

IsInfinity -- Test for an infinity

Standard math library

##### Calling format:

    IsInfinity(expr)

##### Parameters:

`expr` - expression to test

##### Description:

This function tests whether `expr` is an infinity. This is only the case if `expr` is either `Infinity` or `-Infinity`.

##### Examples:

    In> IsInfinity(10^1000);
    Out> False;
    In> IsInfinity(-Infinity);
    Out> True;

##### See also:

Integer .

###

IsPositiveReal -- Test for a numerically positive value

Standard math library

##### Calling format:

    IsPositiveReal(expr)

##### Parameters:

`expr` - expression to test

##### Description:

This function tries to approximate "expr" numerically. It returns `True` if this approximation is positive. In case no approximation can be found, the function returns `False`. Note that round-off errors may cause incorrect results.

##### Examples:

    In> IsPositiveReal(Sin(1)-3/4);
    Out> True;
    In> IsPositiveReal(Sin(1)-6/7);
    Out> False;
    In> IsPositiveReal(Exp(x));
    Out> False;

The last result is because `Exp(x)` cannot be numerically approximated if `x` is not known. Hence Yacas can not determine the sign of this expression.

##### See also:

IsNegativeReal , IsPositiveNumber , N .

###

IsNegativeReal -- Test for a numerically negative value

Standard math library

##### Calling format:

    IsNegativeReal(expr)

##### Parameters:

`expr` - expression to test

##### Description:

This function tries to approximate `expr` numerically. It returns `True` if this approximation is negative. In case no approximation can be found, the function returns `False`. Note that round-off errors may cause incorrect results.

##### Examples:

    In> IsNegativeReal(Sin(1)-3/4);
    Out> False;
    In> IsNegativeReal(Sin(1)-6/7);
    Out> True;
    In> IsNegativeReal(Exp(x));
    Out> False;

The last result is because `Exp(x)` cannot be numerically approximated if `x` is not known. Hence Yacas can not determine the sign of this expression.

##### See also:

IsPositiveReal , IsNegativeNumber , N .

###

IsConstant -- Test for a constant

Standard math library

##### Calling format:

    IsConstant(expr)

##### Parameters:

`expr` - some expression

##### Description:

`IsConstant` returns `True` if the expression is some constant or a function with constant arguments. It does this by checking that no variables are referenced in the expression. `Pi` is considered a constant.

##### Examples:

    In> IsConstant(Cos(x))
    Out> False;
    In> IsConstant(Cos(2))
    Out> True;
    In> IsConstant(Cos(2+x))
    Out> False;

##### See also:

IsNumber , IsInteger , VarList .